Optimizing the Coherence of a Network of Networks

We study the problem of optimal network design in a network of networks, a graph composed of a set of disjoint subgraphs, and a set of designed edges between them. Nodes obey noisy consensus dynamics, and our system model allows for both positive and negative edge weights. We quantify system performance by its coherence, an H-2 norm that captures the steady-state variance of the deviation from consensus. We pose the problem of how to connect the subgraphs, by selecting a single connecting node in each subgraph, so that the resulting network of networks has optimal coherence. We then show that this problem can be solved by identifying the connecting node in each subgraph independently of the other nodes and subgraphs. Thus, the problem can be solved in polynomial time in the order of the largest subgraph. We prove that when the connecting topology is a tree, this solution is optimal even under a more general model that allows for multiple connecting nodes per subgraph. We also derive bounds on the best and worst coherence for a general network of networks with all-positive edge weights. Finally, we provide analytical and numerical examples that further explore coherence in a network of networks.